abstract

This paper discusses the conditions of missile tracking the enemy's trajectory and launching I-type air-to-air missile to destroy the enemy's aircraft, establishes a differential equation model, which is a second-order equation, transforms it into a first-order equation through order reduction, and then solves it by separating variables. The equation expression is obtained, and then the ballistic equation of missile tracking enemy aircraft is obtained. By analyzing the ballistic equations at different speeds, the conditions for launching I-type air-to-air missiles to destroy enemy planes are obtained. Through the process of establishing and solving the model, we get three situations in which the missile tracks the enemy plane:

1，if，，get

At this time, the missile can hit the enemy plane. The time to hit the enemy plane is that the position where the enemy plane was destroyed at that time was

2, if,, get

This time the missile missed the enemy plane.

3, if, at this time, the missile can't hit the enemy plane.

When the flying speed of the enemy plane is v = 1 Mach number, the position is N = 100 km, and the tracking missile speed is U = Mach number 2 (1 Mach number = 1224 km/h, 1 km = 1 km).

When establishing the model, find out the relationship between the known function and the unknown function, and get the expression of the unknown function from one or several listed equations containing the unknown function. According to the tangent property of the curve, the ordinary differential equation is listed, and the expression of the general solution is obtained, from which the special solution needed by the problem can be easily obtained. We can also know the dependence on some parameters from the expression of the general solution, which is convenient for selecting the parameters appropriately, so that the corresponding solution has the required performance, and it is also helpful for other research on the solution.

The main idea is to idealize two objects into particles through the model and turn the problem into a plane problem. According to the corresponding laws, ordinary differential equations are listed, and complex problems are transformed into the form of solving differential equations. In order to obtain the conditions for launching I-type air-to-air missiles to destroy enemy planes, we discuss the most important factor affecting the solution of differential equations-the speed of enemy planes and missiles, so as to solve the problem.

Key words: tangent property of curve, differential equation model, variable separation method, order reduction method.

1, problem restatement:

The radar of our air defense headquarters found an unknown plane. After analyzing and confirming that it was an enemy plane, we ordered our fighter plane patrolling at the same height above the headquarters to launch an I-type air-to-air tracking missile to destroy it (the tracking missile can automatically adjust the tracking direction at any time for the target). Suppose that when the radar finds the enemy plane, it is at an altitude of N kilometers due east of our air defense headquarters, and it wants to escape to a safe area of M kilometers due north at the same altitude (due to the effect of electronic interference, once the enemy plane enters the safe area, the missile loses its tracking target and cannot be destroyed). Under appropriate assumptions, determine the conditions for missiles to track enemy planes and launch I-type air-to-air missiles to destroy enemy planes.

When the flight speed v, position n and tracking missile speed u of the enemy plane are given, the time when the enemy plane was hit and the position where the enemy plane was destroyed at that time are calculated.

2. Assumptions of the model:

2. 1 The missile tracks the enemy plane problem and establishes a two-dimensional coordinate system.

Because our fighter plane and enemy plane are at the same height, and the size of missile and enemy plane is much smaller than their motion range, we regard them as two particles on the same plane and establish xoy rectangular coordinate system.

2.2 Relationship between missile tracking path and intercepting enemy planes

Because the missile can automatically adjust the tracking direction of the target at any time, the connection line between a point on the missile trajectory and a point on the enemy trajectory is tangent to the missile trajectory, and because it is necessary to escape to a safe area in the shortest time, it can be assumed that the enemy trajectory is a straight line. According to the geometric relationship, the function relationship is established, and the trajectory equation of missile tracking enemy aircraft is obtained by differential and integral.

3, the establishment of the model:

The patrol aircraft directly above the headquarters is located at (0,0) point, the positive direction of the X axis is due east, the positive direction of the Y axis is due north, and the trajectory of the enemy aircraft is a straight line X = N. The time is counted from the time when the patrol aircraft launches missiles (at this time, the enemy aircraft is at (n,0) point).

Brief introduction parameters: the missile speed is U Mach number, the enemy plane speed is V Mach number, and the time is t hours. Suppose that after time t, the enemy plane is at point R and the missile is at point D. At this time, the straight line RD is tangent to the missile trajectory, and the tangent point is D. If the missile successfully intercepts the enemy plane, the intersection of the missile trajectory curve and the straight line x=N is just below point (n, m).

The functions of x and y are obtained under different speed conditions. The discussion is divided into five parts.

4, the solution of the model:

Suppose the time after the discovery of the enemy plane, the enemy plane arrives at R(N,), the missile arrives at D (,), and the trajectory of DR is tangent to the missile, which is obtained from the geometric relationship:

( 1)

For elimination, (1) is differentiated to get (2).

The replacement is the displacement of the missile.

(3)

Combining (1) and (2), the differential equation of missile trajectory is obtained.

, (4)

Order, and then (5)

By integrating the two ends of equation (5) and using the initial condition:, we get

, (6)

What kind of function is required must be further determined.

1，if，，integral (6)，get

When the missile hits the enemy plane, the displacement of the enemy plane is

The time spent is

2, if, by (6), get.

Obviously, you can't get n, and the missile can't hit the enemy plane at this time.

If, obviously, the missile can't hit the enemy plane at this time.

5. Result analysis and inspection:

The model transforms the problem into a double-boundary plane problem, which makes the problem simple and clear. The ordinary differential equation is listed by using the tangent property of geometric relationship-curve, and the equation is solved by differential and integral, which simplifies the calculation process and obtains the required results quickly and conveniently.

The model also has some shortcomings. In order to facilitate the establishment of the model and the solution of the problem, we ignore the mass of the missile, so in practical problems, the missile needs to be launched at a certain angle with the horizontal plane. During the movement of missiles and enemy planes, we also ignore the influence of wind speed and direction on their trajectories. So there is a certain error between the obtained data and the actual data.

The model is practical and can also be used for other hunting problems, such as hunting smuggled cargo ships by maritime patrol.

6. Model evaluation:

We list ordinary differential equations according to the corresponding laws. The complex problem is transformed into the form of finding the solution of differential equation, and the dependence on some parameters can be understood from the expression of general solution, which is convenient for selecting parameters appropriately, making the corresponding solution have the required performance, and is also helpful for other research on the solution.

Now, ordinary differential equations have important applications in many disciplines, such as automatic control, the design of various electronic devices, the calculation of trajectory, the study of flight stability of aircraft and missiles, and the study of the stability of chemical reaction process. These problems can be attributed to the solutions of ordinary differential equations, or to the study of the properties of solutions.

7. Reference

Newton's two-body problem: the motion of a single planet under the gravity of the sun. He idealized two objects as particles and got three second-order equations with three unknown functions, which were proved to be plane problems by simple calculation, that is, two second-order differential equations with two unknown functions.

Differential geometry takes smooth curves (surfaces) as the research object, so the whole differential geometry is developed from the concepts of arc length of curves and tangent of a point on curves. Because differential geometry studies the properties of general curves and surfaces, the curvature of a plane curve at one point and the curvature of a space curve at one point are important discussions in differential geometry, and it is necessary to calculate the curvature of each point on a curve or surface by differential method.